March 26, 2013

Assuming you have a standard Gaussian bell curve (in blue). Suppose that you want to cut it into two parts of equal areas, with an horizontal line.

Which fraction of the Gaussian peak height provides you with the red and the green curves, which sum up to the Gaussian, with equal surface integral (undr the red and the green cuves)?

It turns out that, numerically, the fraction, on the y-axis, is about 0.3063622804625085, or one over 3.26410940175247 of the peak height.

If one looks at the x-axis, one has to cut at +/- 1.538172262286592.\sigma, where \sigma is the usual Gaussian scale parameter. In practice, cutting the Gaussian at 3/10 of the height would be good enough, assuming sufficient, far to critical, sampling. Yet out of curiosity, i looked at several numerical constant tables, or even Plouffe's constant inverter, and did not find any of these three. So once again, the potential gaussian split constants are:

0.3063622804625085

1.538172262286592

3.26410940175247

Does anybody knows whether this Gaussian split is "common practice" in some mathematical field, or if these constants are listed somewhere?Though application dwells in the realm of fast Gaussian filter approximation. More to come.

Abstract: The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share a hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”. We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.